TheoremStatement: AI-adaptedProof: AI-generatedverified 2026-09-26 (gpt-6-sol)

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Extension-theoretic interpretation of the standard five-term exact sequence

Statement

Assume the Axiom of Choice (The Axiom of Choice). For an extension $1 \to N \to G \to Q \to 1$ and an abelian $G$ -module $A$ , assume the standard low-degree sequence

0 \to H^{1} (Q, A^{N}) \overset{Inf}{\to} H^{1} (G, A) \overset{Res}{\to} H^{1} (N, A)^{Q} \overset{Tra}{\to} H^{2} (Q, A^{N}) \overset{Inf}{\to} H^{2} (G, A)

is exact. Then the transgression detects extension of degree-one classes: for $[u] \in H^{1} (N, A)^{Q}$ one has $Tra [u] = 0$ exactly when $[u]$ is the restriction of a class in $H^{1} (G, A)$ . Under the factor-set classification, the last inflation map is represented by pulling a $Q$ -extension back along $G \to Q$ and then pushing it out along the inclusion $A^{N} ↪ A$ .

Facts & Assumptions

Given: The Axiom of Choice, an extension $1 \to N \to G \to Q \to 1$ , and an abelian $G$ -module $A$ .

[A2]

Choice supplies normalized sections to apply the extension classification [L2].

[F1]

Restriction and inflation in degree one are the explicit maps defined on cocycles in Restriction, inflation, and the quotient conjugation action on first cohomology.

[L1]

The degree-one inflation-restriction sequence is exact (Inflation-restriction exact sequence in degree one).

[L2]

$H^{2} (Q, A^{N})$ classifies extensions of $Q$ by $A^{N}$ (H^2 classifies extensions with fixed abelian kernel action).

[A1]

The displayed five-term sequence is the standard exact low-degree five-term sequence attached to $1 \to N \to G \to Q \to 1$ .

Proof

technique · direct

1.1F1L1given

The first three terms are exactly the degree-one inflation-restriction sequence, so they are exact by [L1], with maps described concretely by [F1].

1.2A2L2givenalgebra

Under [A2] and [L2], a class in $H^{2} (Q, A^{N})$ is an extension of $Q$ by $A^{N}$ . The map to $H^{2} (G, A)$ first pulls that extension back along $G \to Q$ , producing an extension of $G$ by $A^{N}$ , and then pushes out along the inclusion $A^{N} ↪ A$ . That is the extension-theoretic meaning of the last inflation map in the displayed sequence.

1.3A1algebra

Exactness of [A1] at $H^{1} (N, A)^{Q}$ says $\ker (Tra) = im (Res) .$ Thus $Tra [u] = 0$ exactly when $[u]$ is the restriction of a degree-one class on $G$ , which is precisely the asserted extension criterion for the cohomology class.

2.1step 1.1step 1.2step 1.3∎

Steps 1.2 and 1.3 give the two claimed interpretations, while step 1.1 identifies the preceding degree-one maps.

Depends on

Used by

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Sources

Clara Loh, Group Cohomology, SS 2019 (standard reference, not scraped)
Caroline Lassueur, Cohomology of Groups, SS 2021 (standard reference, not scraped)